Magnetic exchange coupling energy calculating method and apparatus

ABSTRACT

A non-transitory computer-readable recording medium stores a magnetic program causing a computer to perform an exchange coupling energy calculating process including interpolating a rotation angle between two magnetization vectors disposed at the respective centers of two adjacent elements used in a finite volume method with reference to a rotation axis perpendicular to the two magnetization vectors, and calculating a magnetic exchange coupling field by integrating a magnetic field acting as a force exerted on the two magnetization vectors with the interpolated rotation angle.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is based upon and claims the benefit of priority ofJapanese Patent Application 2010-174011, filed on Aug. 2, 2010, theentire contents of which are incorporated herein by reference.

FIELD

An embodiment of the invention discussed herein relates to a magneticexchange coupling energy calculating method and apparatus.

BACKGROUND

According to the related art, an apparatus calculates an averagemagnetic flux density and an average magnetic field in an equivalentelement occupied by plural substances including a magnetic material, andanalyzes an electromagnetic field produced in a area larger than theequivalent element based on the calculated average magnetic flux densityand average magnetic field. A system is also known in which a targetarea for electromagnetic field analysis is divided into micro-areas, andan H-B curve and a W-B curve each having a ratio α of the minimummagnetic flux density to the maximum magnetic flux density in themicro-areas as a parameter are stored in a data base in order to performanalysis of an electromagnetic field.

In another apparatus according to the related art, the structure of amagnetic domain, which is an area in which the directions of magneticmoments of atoms are oriented in the same direction, in a magneticmaterial is varied in a stepwise manner. The apparatus calculates themagnetic energy of the magnetic material in each step, and determinesthe step in which the magnetic energy is minimized. A technology is alsoknown by which micro-magnetization analysis is performed by taking intoconsideration magnetic properties, such as magnetic anisotropy, inaccordance with a program.

Micro-magnetization analysis refers to a technique for modeling amagnetic material, such as a magnetic head of a HDD (Hard Disk Drive),as a collection of small magnets, as illustrated in FIG. 1, in order tosimulate the state of magnetic domains numerically.“Micro-magnetization” refers to an individual small magnet or magneticmaterial element. In micro-magnetization analysis, a mesh of about 10 nmmay be used instead of a mesh of a size corresponding to the actualatomic-size order, from the viewpoint of calculation cost. In a typicalmesh size (such as 10 nm or less), the magnetization vectors in adjacentmesh areas may form angles of 5° or less, so that the angles may beconsidered to be substantially continuous.

The motion of micro-magnetization is governed by a governing equationreferred to as the LLG (Landau-Lifshitz-Gilbert) equation, asillustrated below:

$\begin{matrix}{\frac{\partial\overset{\rightarrow}{M}}{\partial t} = {{{- \gamma}\;\overset{\rightarrow}{M} \times {\overset{\rightarrow}{H}}_{eff}} + {\alpha\left( {\overset{\rightarrow}{M} \times \frac{\partial\overset{\rightarrow}{M}}{\partial t}} \right)}}} & (1)\end{matrix}$where M, γ, α, and H_(eff) are a magnetization vector, a magneticrotation ratio, a frictional coefficient, and an effective magneticfield, respectively.

The effective magnetic field H_(eff) is a composition of plural magneticfield vectors, as indicated by Equation (2) below. The magnetic fieldsto which the micro-magnetization is subject include an external magneticfield H_(out), a demagnetizing field H_(demag), an anisotropic magneticfield H_(an), and a magnetic exchange coupling field H_(ex).{right arrow over (H)} _(eff) ={right arrow over (H)} _(out) +{rightarrow over (H)} _(demag) +{right arrow over (H)} _(an) +{right arrowover (H)} _(ex)  (2)

The magnetic exchange coupling field H_(ex) exerts a force thatoriginally acts between adjacent atoms. In order to performmicro-magnetization analysis by using an analysis model in which thesize of the mesh is larger than the inter-atomic distance whilemaintaining calculation accuracy, an analysis model may be adopted inwhich the size of the mesh is so small (such as 10 nm or less) that theangles of adjacent magnetization vectors vary by 10 degrees or less.

Micro-magnetization analysis has been mainly used for small amounts ofmagnetic material of micron order as an analysis target. However, due toadvances in computing technology, it is now possible to applymicro-magnetization analysis for magnetic materials of several dozenmicron order. It is expected that magnetic material areas of evengreater sizes, such as those of motors and transformers, will beselected as analysis targets.

Patent Document 1: Japanese Laid-open Patent Publication No. 2005-43340

Patent Document 2: Japanese Laid-open Patent Publication No. 2004-347482

Patent Document 3: Japanese Laid-open Patent Publication No. 2004-219178

Patent Document 4: Japanese Laid-open Patent Publication No. 2005-100067

Thus, the mesh size may preferably be 10 nm or less in order to enablehighly accurate micro-magnetization analysis. As the size of theanalysis target area is increased, the number of meshes (which may behereafter referred to as the number of mesh areas) that need to behandled increases. An increase in mesh size decreases the degrees offreedom required for calculation, so that calculation time can bereduced. However, when the mesh size is increased, the adjacentmagnetization vectors may form an angle of more than 10°. Thus, simplyincreasing the mesh size may result in a decrease in calculationaccuracy.

Further, in a micro-magnetization analysis according to related art, anincrease in mesh size (such as more than 10 nm) results in an increasein the rotation angle of the magnetization vectors (angle formed byadjacent magnetization vectors), thus preventing accuratemicro-magnetization analysis. Specifically, the accuracy of calculationof the magnetic exchange coupling energy or the magnetic exchangecoupling field may be lowered.

The object and advantages of the disclosure will be realized andattained by means of the elements and combinations particularly pointedout in the claims.

It is to be understood that both the foregoing general description andthe following detailed description are exemplary and explanatory and arenot restrictive of the invention, as claimed.

SUMMARY

According to an aspect of the invention, a non-transitorycomputer-readable recording medium stores a program causing a computerto perform a magnetic exchange coupling energy calculating processincluding interpolating a rotation angle between two magnetizationvectors disposed at the respective centers of two adjacent elements usedin a finite volume method with reference to a rotation axisperpendicular to the two magnetization vectors, and calculating amagnetic exchange coupling field by integrating a magnetic field actingas a force exerted on the two magnetization vectors with theinterpolated rotation angle.

BRIEF DESCRIPTION OF DRAWINGS

FIG. 1 illustrates an example of a magnetic material model used in amicro-magnetization analysis according to the related art;

FIG. 2 is a block diagram of a magnetic exchange coupling energycalculating apparatus according to an embodiment of the presentinvention;

FIG. 3 illustrates an example where two magnetization vectors form anangle of 90°;

FIG. 4 is a graph plotting an error between the magnetic exchangecoupling energy according to equation (4) and that according to equation(5), relative to a rotation angle of magnetization vectors;

FIG. 5 illustrates two magnetization vectors disposed according to afinite volume method;

FIG. 6 illustrates a volume occupied by the two adjacent magnetizationvectors;

FIG. 7 illustrates a rotation axis u perpendicular to the twomagnetization vectors, and a rotation angle θ;

FIG. 8 illustrates an example where the angle formed by the twomagnetization vectors is interpolated, and the interval between the twomagnetization vectors is divided into N portions;

FIG. 9 is a flowchart of a process of calculating magnetic exchangecoupling energy;

FIG. 10 is a flowchart of a process of calculating a magnetic exchangecoupling field;

FIG. 11 illustrates a solution of magnetic domain state;

FIGS. 12A, 12B, and 12C illustrate results of a micro-magnetizationanalysis in which a magnetic material model is divided by a mesh size of10 nm; and

FIGS. 13A, 13B, and 13C illustrate results of a micro-magnetizationanalysis in which the magnetic material model is divided by a mesh sizeof 30 nm.

DESCRIPTION OF EMBODIMENTS

Embodiments of the present invention will be described with reference tothe accompanying drawings. FIG. 2 is a block diagram of a computer 1 asan example of a magnetic exchange coupling energy calculating apparatusaccording to an embodiment. The computer 1 includes a CPU (CentralProcessing Unit) 11 that controls the overall operation of the computer1. The CPU 11 may provide an interpolating function and a calculatingfunction. The computer 1 also includes a memory 12 that may provide aworking area; a hard disk drive (HDD) 13 that may store an operatingsystem (OS) and a simulation program; a network interface card (NIC) 14;an input interface (I/F) 15; and a video I/F 16.

The CPU 11 is connected to the memory 12, the HDD 13, the NIC 14, theinput I/F 15, and the video I/F 16 via a bus or the like. To the inputI/F 15, a keyboard and mouse 17 are connected. To the video I/F 16, amonitor 18 is connected. The CPU 11 may read the simulation program fromthe HDD 13 and execute it in order to perform micro-magnetizationanalysis utilizing a finite volume method. Data of a magnetic materialmodel used in micro-magnetization analysis may be stored in the HDD 13in advance. The simulation program may be initially recorded in anon-transitory computer-readable recording medium 20 and later loadedinto the HDD 13.

Next, the problem of a decrease in accuracy of calculation of themagnetic exchange coupling energy is discussed with reference to asimple calculation based on two magnetization vectors. When the LLGequation of magnetization vectors according to equation (1) isdiscretized by the simulation program, the magnetic exchange couplingenergy may be expressed by the following:

$\begin{matrix}\begin{matrix}{E_{ij} = {\int{e{\mathbb{d}V}}}} \\{= {- {\int{\frac{A}{M_{S}L^{2}}\left( {{\overset{\rightarrow}{M}}_{j} - {\overset{\rightarrow}{M}}_{i}} \right)^{2}{\mathbb{d}V}}}}} \\{= {{- \frac{A{\mathbb{d}S}}{M_{S}L^{2}}}{\int_{0}^{L}{\left( {{\overset{\rightarrow}{M}}_{j} - {\overset{\rightarrow}{M}}_{i}} \right)^{2}\ {\mathbb{d}l}}}}}\end{matrix} & (3)\end{matrix}$where e is a magnetic exchange coupling energy density; A is a magneticexchange coupling coefficient, M is saturation magnetization per unitvolume of magnetic material; L is the distance between adjacentmagnetization vectors; M_(i) is the i-th magnetization vector; dS is asectional area between adjacent mesh areas; and θ is an angle formed bythe adjacent magnetization vectors.

When the angle formed by the two magnetization vectors is 90°, asillustrated in FIG. 3, the LLG equation of magnetization vectorsaccording to equation (1) may be discretized in accordance with thesimulation program by a related-art technique such that themagnetization vectors are discontinuous between two mesh areas. In thiscase, the magnetic exchange coupling energy may be expressed by thefollowing:

$\begin{matrix}{\left\lbrack {{METHOD}\mspace{14mu}{BY}\mspace{14mu}{DISCRETIZATION}} \right\rbrack\mspace{14mu}\begin{matrix}{E_{ij} = {{- \frac{A{\mathbb{d}S}}{M_{S}^{2}L^{2}}}{\int_{0}^{L}{\left( {{\overset{\rightarrow}{M}}_{j} - {\overset{\rightarrow}{M}}_{i}} \right)^{2}\ {\mathbb{d}l}}}}} \\{= {{- \frac{A{\mathbb{d}S}}{M_{S}^{2}L}}\left( {{\overset{\rightarrow}{M}}_{j} - {\overset{\rightarrow}{M}}_{i}} \right)^{2}}} \\{= {{- \frac{A{\mathbb{d}S}}{L}}\left( {2 - {2\;\cos\;\theta}} \right)}} \\{= {- \frac{2A{\mathbb{d}S}}{L}}}\end{matrix}} & (4)\end{matrix}$

Similarly, when the angle formed by the two magnetization vector is 90°,the magnetic exchange coupling energy may be calculated in accordancewith the simulation program by integrating magnetization vectors thathave been interpolated between two mesh areas. In this case, themagnetic exchange coupling energy may be expressed by equation (5). Thiscalculation formula may be analytically obtained by integrating the areaoccupied by the two magnetization vectors by the angle θ, on theassumption that the rotation angle of the two magnetization vectors islinearly and continuously varied.

$\begin{matrix}{\left\lbrack {{ANALYTIC}\mspace{14mu}{METHOD}} \right\rbrack\begin{matrix}{E_{ij} = {{- \frac{A{\mathbb{d}S}}{M_{S}^{2}L^{2}}}{\int_{0}^{L}{\left( {{\overset{\rightarrow}{M}}_{j} - {\overset{\rightarrow}{M}}_{i}} \right)^{2}\ {\mathbb{d}l}}}}} \\{= {{- \frac{A{\mathbb{d}S}}{M_{S}^{2}{L\left( {\pi/2} \right)}}}{\int_{0}^{\pi/2}{\left( {2 - {2\;\cos\;\Theta}} \right)\ {\mathbb{d}\theta}}}}} \\{= {- {\frac{2A{\mathbb{d}S}}{M_{S}^{2}L\;\pi}\left\lbrack {{2\;\theta} - {2\;\sin\;\theta}} \right\rbrack}_{0}^{\pi/2}}} \\{= {{- \frac{2A{\mathbb{d}S}}{L}}\left( {1 - \frac{2}{\pi}} \right)}}\end{matrix}} & (5)\end{matrix}$

The magnetic exchange coupling energy of the former, i.e., equation (4),is lower by about 20% in absolute values than the magnetic exchangecoupling energy of the latter, i.e., equation (5). Namely, when therotation angle of the magnetization vectors is large, such as 90°, anaccurate magnetic exchange coupling energy cannot be obtained by therelated-art technique.

FIG. 4 is a graph plotting the error between the magnetic exchangecoupling energy according to equation (4) and that according to equation(5), relative to the rotation angle of the magnetization vectors. Asillustrated in FIG. 4, the error increases as the rotation angle of themagnetization vectors increases. Namely, when the mesh size is increased(to more than 10 nm, for example), the rotation angle of themagnetization vectors also increases, and therefore the error inmagnetic exchange coupling energy also increases.

In accordance with the present embodiment, even when a magnetic materialmodel is created that includes a mesh of a size (such as 30 nm) largerthan those of related-art meshes, the area occupied by the twomagnetization vectors is integrated (analytical solution) in accordancewith the simulation program on the assumption that the rotation angle ofthe magnetization vectors of the adjacent mesh areas is continuously andlinearly varied, thus enabling an accurate calculation of magneticexchange coupling energy.

Next, a method of accurately calculating the magnetic exchange couplingenergy is described in detail. Generally, a discretizing method formicro-magnetization analysis may involve the finite volume method,whereby a magnetization vector is disposed at the center of an element.Although the finite volume method does not involve interpolationfunctions such as those of the finite element method, an elementboundary surface value may be obtained by linearly interpolating thevalue at the center of the element. FIG. 5 illustrates magnetizationvectors M_(i) and M_(j) disposed at the respective centers of twoadjacent mesh areas A and B (elements) according to the finite volumemethod. In the following description, the magnetic exchange couplingenergy is accurately determined by applying the concept of linearinterpolation to the finite volume method in accordance with thesimulation program, on the assumption that the rotation angle of themagnetization vectors of adjacent mesh areas is continuously varied.

Referring to FIG. 6, the volume V_(ij) occupied by the two adjacentmagnetization vectors is illustrated by the area with hatching. Themagnetic exchange coupling energy of the area is expressed by thefollowing:

$\begin{matrix}\begin{matrix}{E_{ij} = {\int{e{\mathbb{d}V_{ij}}}}} \\{= {- {\int{\frac{A}{M_{S}L_{ij}^{2}}\left( {{\overset{\rightarrow}{M}}_{j} - {\overset{\rightarrow}{M}}_{i}} \right)^{2}{\mathbb{d}V_{ij}}}}}} \\{= {- {\int_{0}^{L}{\frac{A}{M_{S}L_{ij}^{2}}\left( {{\overset{\rightarrow}{M}}_{j} - {\overset{\rightarrow}{M}}_{i}} \right)^{2}{\mathbb{d}S_{ij}}{\mathbb{d}l_{ij}}}}}}\end{matrix} & (6)\end{matrix}$where dS_(ij) is a sectional area between the two mesh areas, and L_(ij)is the distance between the two magnetization vectors. dV_(ij) isexpressed by the following equation (7), while L_(ij) is expressed bythe following equation (8):dV _(ij) =dS _(ij) dl _(ij)(dl _(ij) =d{right arrow over (x)} _(ij) ·dn_(ij))  (7)L _(ij) ={right arrow over (x)} _(ij) ·{right arrow over (n)} _(ij)  (8)

Equation (9) is a related-art equation indicating the magnetic exchangecoupling energy between the two magnetization vectors M_(i) and M_(i).While equation (9) may be used when the rotation angle of the adjacentmagnetization vectors is small (such as 10° or less), equation (9) isassociated with the problem of an increase in error in magnetic exchangecoupling energy as the rotation angle of the adjacent magnetizationvectors is increased.

$\begin{matrix}{\left\lbrack {{RELATED}\;\text{-}{ART}\mspace{14mu}{METHOD}} \right\rbrack{E_{ij} = {{- \frac{A{\mathbb{d}S_{ij}}}{M_{S}L_{ij}}}\left( {{\overset{\rightarrow}{M}}_{j} - {\overset{\rightarrow}{M}}_{i}} \right)^{2}}}} & (9)\end{matrix}$

The magnetic exchange coupling energy may be accurately calculated bycalculating the angle formed by the two magnetization vectors M_(i) andM_(j), integrating the energy of the volume occupied by the twomagnetization vectors M_(i) and M_(j) by using a linearly interpolatedrotation angle, and analytically determining the magnetic exchangecoupling energy in accordance with the simulation program. Because therotation angle is linearly varied, when the angle θ formed by the twomagnetization vectors is equal to or less than π, the rotation axis uand the rotation angle θ illustrated in FIG. 7 can be calculated by thefollowing equations (10) and (11), where M_(s) is the magnitude of themagnetization vector. The rotation axis u corresponds to a vectorperpendicular to the two magnetization vectors M_(i) and M_(j).

$\begin{matrix}{u = {\left( {u_{x},u_{y},u_{z}} \right) = \frac{{\overset{->}{M}}_{i} - {\overset{->}{M}}_{j}}{M_{s}^{2}}}} & (10) \\{\frac{{\overset{->}{M}}_{i} - {\overset{->}{M}}_{j}}{M_{s}^{2}} = {\left. {\cos\;\Theta_{ij}}\Rightarrow\Theta_{ij} \right. = {{\arccos\left( \frac{{\overset{->}{M}}_{i} - {\overset{->}{M}}_{j}}{M_{s}^{2}} \right)}\mspace{14mu}\left( {0 \leq \theta \leq \pi} \right)}}} & (11)\end{matrix}$

When the integral of distance in one dimension of the magnetic exchangecoupling energy is variably transformed into the integral of rotationangle, equation (6) may be transformed into equation (12):

$\begin{matrix}\begin{matrix}{{{- \frac{A}{M_{S}L^{2}}}{\int_{0}^{L}{\left( {{\overset{->}{M}}_{j} - {\overset{->}{M}}_{i}} \right)^{2}{\mathbb{d}S_{ij}}{\mathbb{d}l_{ij}}}}} = {{- \frac{A\;{\mathbb{d}S_{ij}}}{M_{S}L^{2}}}{\int_{0}^{L}{\left( {{\overset{->}{M}}_{j} - {\overset{->}{M}}_{i}} \right)^{2}{\mathbb{d}l_{ij}}}}}} \\{= {{- \frac{A\;{\mathbb{d}S_{ij}}}{M_{S}L^{2}}}{\int_{0}^{L}{\begin{pmatrix}{{\overset{->}{M}}_{j}^{2} + {\overset{->}{M}}_{i}^{2} -} \\{2{{\overset{->}{M}}_{j} \cdot {\overset{->}{M}}_{i}}}\end{pmatrix}{\mathbb{d}l_{ij}}}}}} \\{= {{- \frac{A\; L{\mathbb{d}S_{ij}}}{L^{2}\Theta_{ij}}}{\int_{0}^{\Theta}{\left( {2 - {2\cos\;\theta}} \right){\mathbb{d}\theta}}}}} \\{= {{- \frac{2A\;{\mathbb{d}S_{ij}}}{L\;\Theta_{ij}}}{\int_{0}^{\Theta}{\left( {1 - {\cos\;\theta}} \right){\mathbb{d}\theta}}}}} \\{\left( {{\mathbb{d}l_{ij}} = {\frac{L_{ij}}{\Theta_{ij}}{\mathbb{d}\theta_{ij}}}} \right)}\end{matrix} & (12)\end{matrix}$where, when the integral with respect to the angle θ on the right-handside is analytically calculated in accordance with the simulationprogram, accurate magnetic exchange coupling energy is expressed by thefollowing:

$\begin{matrix}{{\left\lbrack {{ANALYTIC}\mspace{14mu}{METHOD}} \right\rbrack\mspace{14mu} E_{ij}} = {{- \frac{2A\;{\mathbb{d}S_{ij}}}{L_{ij}\Theta}}\left( {\Theta - {\sin\;\Theta}} \right)}} & (13)\end{matrix}$

Equation (13) indicates the result of analytically calculating themagnetic exchange coupling energy in accordance with the simulationprogram. It is also possible to accurately calculate the magneticexchange coupling energy by interpolating the angle formed by themagnetization vectors M_(i) and M_(j), dividing the integral interval ofequation (6) into N portions, and numerically integrating the energy ofthe volume occupied by the magnetization vectors M_(i) and M_(j) inaccordance with the simulation program, as illustrated in FIG. 8.Equation (6), when expressed in the form of numerical integration, maybe expressed by equation (14), where M₁ corresponds to M_(i) and M_(N)corresponds to M_(j); w is a weighting function of a Gaussian integral;and N is the number of divisions of the integral interval, which may bedesignated by a user.

$\begin{matrix}{{\left\lbrack {{NUMERICAL}\mspace{14mu}{INTEGRATION}\mspace{14mu}{METHOD}} \right\rbrack\mspace{14mu} E_{ij}} = {{- \frac{A\;{dS}_{ij}}{M_{S}L_{ij}N}}{\sum\limits_{n = 1}^{N}{w_{i}\left( {{\overset{->}{M}}_{n} - {\overset{->}{M}}_{n + 1}} \right)}^{2}}}} & (14)\end{matrix}$

FIG. 9 is a flowchart of a process of calculating the magnetic exchangecoupling energy in accordance with the simulation program. First, themagnetic exchange coupling energy is initialized; namely, the exchangecoupling energy E_(ex) is set to 0 (step S1). Next, the value of “i” ofthe magnetization vector M_(i) as a calculation target included in themagnetic material model is sequentially incremented from “1” to thetotal number of mesh areas (the loop between steps S2-1 and S2-2). Thetiming of incrementing the value of “i” corresponds to the timing ofstep S2-2 when the processes of steps S4 and S5 are completed, as willbe described later. When the value of “i” of the magnetization vectorM_(i) is set equal to the total number of mesh areas, and the processesof steps S4 and S5 are completed, the calculation process is completed.

In accordance with the simulation program, the value of “j” of themagnetization vector M_(j) adjacent to the magnetization vector M_(i) asa calculation target included in the magnetic material model issuccessively incremented from “0” to the total number of mesh areas-1(the loop between steps S3-1 and S3-2). The timing of incrementing thevalue of “j” corresponds the timing of step S3-2 when the processes ofsteps S4 and S5 are completed, as will be described later.

After step S3-1, the volume dV_(ij) occupied by the two magnetizationvectors M_(i) and M_(j) and the distance L_(ij) between the twomagnetization vectors M_(i) and M_(j) are calculated in accordance withequations (7) and (8). Also, the rotation angle θ is calculated inaccordance with equation (11) (step S4). Thereafter, the magneticexchange coupling energy E_(ij) is calculated based on the calculatedvolume dV_(ij), distance L_(ij), and rotation angle θ according toequation (13). The calculated magnetic exchange coupling energy E_(ij)is added to the preceding total of the magnetic exchange coupling energyE_(ex), thus obtaining the total magnetic exchange coupling energyE_(ex) (step S5). In step S5, the magnetic exchange coupling energyE_(ij) is calculated each time the values of “i” or “j” of themagnetization vectors M_(i) and M_(j) are incremented.

Preferably, the simulation program may be modified such that themagnetic exchange coupling energy E_(ij) is calculated according toequation (14) in step S5. In accordance with the simulation program, themagnetic exchange coupling field may be calculated by differentiatingthe magnetic exchange coupling energy E_(ij) with the magnetizationvector (H_(ex,ij)=−∂E_(ij)/∂M_(i)).

Thus, in accordance with the simulation program, the rotation anglebetween the two magnetization vectors M_(i) and M_(j) disposed at therespective centers of the adjacent elements used in the finite volumemethod is linearly interpolated with reference to the rotation axis uperpendicular to the two magnetization vectors M_(i) and M_(j), and theenergy of the volume V occupied by the two magnetization vectors M_(i)and M_(j) is integrated by the linearly interpolated rotation angle,thus calculating the magnetic exchange coupling energy. Thus, themagnetic exchange coupling energy can be calculated more accurately thanis possible with related-art methods. Thus, the computer 1 may act as aninterpolating unit and a magnetic exchange coupling energy calculatingunit by executing the simulation program.

Preferably, the simulation program may be modified such that therotation angle between the two magnetization vectors M_(i) and M_(j) isinterpolated, the interval between the two magnetization vectors M_(i)and M_(j) is divided into plural areas, and the energy of the volume Voccupied by the two magnetization vectors M_(i) and M_(j) is numericallyintegrated for each divided area. In this way, the magnetic exchangecoupling energy can be calculated more accurately than is possible withrelated-art methods.

Preferably, the simulation program may be modified such that themagnetic exchange coupling field obtained by differentiating thecalculated magnetic exchange coupling energy with the magnetizationvector is used as the magnetic exchange coupling field included in theeffective magnetic field of the LLG equation, so that a highly accuratemicro-magnetization analysis can be realized.

Next, a method of accurately calculating the magnetic exchange couplingfield is described. In the LLG equation of micro-magnetization, themagnetic field due to a magnetic exchange coupling that acts as a forceexerted on micro-magnetization vectors is expressed by equation (15):

$\begin{matrix}\begin{matrix}{{\overset{->}{H}}_{{ex},{ij}} = {- \frac{\partial E_{ij}}{\partial{\overset{->}{M}}_{i}}}} \\{= {\int{\frac{2A}{M_{S}L^{2}}\left( {{\overset{->}{M}}_{j} - {\overset{->}{M}}_{i}} \right){\mathbb{d}V_{ij}}}}} \\{= {\frac{2A{\mathbb{d}S_{ij}}}{M_{S}L^{2}}{\int_{0}^{L}{\left( {{\overset{->}{M}}_{j} - {\overset{->}{M}}_{i}} \right){\mathbb{d}l_{ij}}}}}}\end{matrix} & (15)\end{matrix}$

In a related-art method, the magnetic exchange coupling field may becalculated according to equation (16) based on a calculationapproximating dl_(ij) as L:

$\begin{matrix}{{\left\lbrack {{RELATED}\text{-}{ART}\mspace{14mu}{METHOD}} \right\rbrack\mspace{14mu}{\overset{->}{H}}_{{ex},{ij}}} \approx {\frac{2A\;{\mathbb{d}S_{ij}}}{M_{S}L}\left( {{\overset{->}{M}}_{j} - {\overset{->}{M}}_{i}} \right)}} & (16)\end{matrix}$

Such an approximate calculation is associated with the problem that asthe rotation angle of the magnetization vectors increases, thecalculation accuracy decreases. In order to more accurately calculatethe magnetic field due to magnetic exchange coupling, integration may bepreferably performed analytically in accordance with the simulationprogram.

According to the analytical calculation according to equation (15),integration is performed with the angle θ of rotation about the rotationaxis of the two magnetization vectors M_(i) and M_(j). The rotation axisu perpendicular to the two magnetization vectors M_(i) and M_(j) isexpressed by equation (17) using a vector product. By rotating themagnetization vector M_(i) about the rotation axis u, the magnetizationvector M_(j) can be expressed as a function of the rotation angle, sothat analytical integration can be performed based on the rotationangle. A transform matrix T(M_(i)→M_(j)) that rotates the magnetizationvector M_(i) about the rotation axis u by θ is expressed by equation(18), where I_(ij) is the unit matrix, while a transform equation forthe magnetization vector M_(j) is expressed by equation (19):

$\begin{matrix}{u = {\frac{{\overset{->}{M}}_{i} \times {\overset{->}{M}}_{j}}{M_{S}^{2}} = \left( {u_{x},u_{y},u_{z}} \right)}} & (17) \\{{T_{ij}(\theta)} = {{I_{ij}\cos\;\theta} + {\left( {1 - {\cos\;\theta}} \right)u_{i}u_{j}} + {\sin\;{\theta\begin{pmatrix}0 & {- u_{z}} & u_{y} \\u_{z} & 0 & {- u_{x}} \\{- u_{y}} & u_{x} & 0\end{pmatrix}}}}} & (18)\end{matrix}${right arrow over (M)} _(j) =T _(ij)(θ){right arrow over (M)} _(i)  (19)

The amount of change in the magnetization vectors, when expressed by atransform matrix T, is expressed by equation (20):

$\begin{matrix}\begin{matrix}{{{\overset{->}{M}}_{j} - {\overset{->}{M}}_{i}} = {\left( {{T_{ij}(\theta)} - 1} \right)M_{i}}} \\{= {\begin{Bmatrix}{{I_{j}\left( {{\cos\;\theta} - 1} \right)} + {\left( {1 - {\cos\;\theta}} \right)u_{i}u_{j}} +} \\{\sin\;{\theta\begin{pmatrix}0 & {- u_{z}} & u_{y} \\u_{z} & 0 & {- u_{x}} \\{- u_{y}} & u_{x} & 0\end{pmatrix}}}\end{Bmatrix}{\overset{->}{M_{i}}.}}}\end{matrix} & (20)\end{matrix}$

When, in accordance with the simulation program, the magnetic exchangecoupling field of equation (15) is analytically integrated by usingequation (20), equation (21) is obtained, where, in accordance with thesimulation program, the integral of distance in one dimension of themagnetic exchange coupling field according to equation (15) is variablytransformed into the integral of the rotation angle θ.

$\begin{matrix}\begin{matrix}{{\left\lbrack {{ANALYTIC}\mspace{14mu}{METHOD}} \right\rbrack\mspace{14mu}{\overset{->}{H}}_{{ex},{ij}}} = {\frac{2A\;{\mathbb{d}S_{ij}}}{M_{S}L\;\Theta_{ij}}{\overset{->}{M}}_{i}}} \\{\int_{0}^{\Theta}{\left( {{T_{ij}(\theta)} - 1} \right){\mathbb{d}\theta}}} \\{= {\frac{2A\;{\mathbb{d}S_{ij}}}{M_{S}L\;\Theta_{ij}}{\tau_{ij}\left( \Theta_{ij} \right)}\overset{->}{M_{i}}}}\end{matrix} & (21)\end{matrix}$where τ_(ij) is expressed by equation (22):

$\begin{matrix}{{\tau_{ij}\left( \Theta_{ij} \right)} = \begin{Bmatrix}{{I_{ij}\left( {{\sin\;\Theta_{ij}} - \Theta_{ij}} \right)} + {\left( {{{- \sin}\;\Theta_{ij}} + \Theta_{ij}} \right)u_{i}u_{j}} +} \\{\left( {1 - {\cos\;\Theta_{ij}}} \right)\begin{pmatrix}0 & {- u_{z}} & u_{y} \\u_{z} & 0 & {- u_{x}} \\{- u_{y}} & u_{x} & 0\end{pmatrix}}\end{Bmatrix}} & (22)\end{matrix}$

The magnetic exchange coupling field can be more accurately calculatedby using equations (21) and (22) in accordance with the simulationprogram than is possible with the related-art equation (16).

Equations (21) and (22) indicate the result of analytically integratingthe magnetic exchange coupling field. Alternatively, the magneticexchange coupling field may also be accurately calculated by modifyingthe simulation program such that the angle formed by the magnetizationvectors M_(i) and M_(j) is interpolated, the integral interval ofequation (15) is divided into N portions, and the magnetic field thatacts as a force exerted on the micro-magnetization vectors isnumerically integrated. Equation (15), when expressed in the form ofnumerical integration, may yield equation (23), where M₁ corresponds toM_(i), M_(N) corresponds to M_(j), and w indicates a weighting functionof a Gaussian integral. The number N of divisions of the integralinterval may be designated by the user.

$\begin{matrix}{{\overset{->}{H}}_{{ex},{ij}} = {\frac{2A\;{\mathbb{d}S_{ij}}}{M_{S}L_{ij}N}{\sum\limits_{n = 1}^{N - 1}{w_{i}\left( {{\overset{->}{M}}_{n} - {\overset{->}{M}}_{n + 1}} \right)}}}} & (23)\end{matrix}$

In accordance with the simulation program, highly accuratemicro-magnetization analysis can be performed by using the value ofequation (21), which is an analytical solution, or the value of equation(23), which is obtained by numerical integration, as the magneticexchange coupling field; namely the effective magnetic field of the LLGequation.

FIG. 10 is a flowchart of a process of calculating the magnetic exchangecoupling field according to the simulation program. First, in accordancewith the simulation program, the magnetic exchange coupling field isinitialized; namely, the magnetic exchange coupling field H_(ex,ij) isset to 0 (step S11). Then, the value of “i” of the magnetization vectorM_(i) as a calculation target included in the magnetic material model issuccessively incremented from “1” to the total number of mesh areas (theloop between steps S12-1 and S12-2). The timing of incrementing thevalue of “i” corresponds to the timing of S12-2 when the processes ofsteps S14 and S15 are completed. When the value of “i” of themagnetization vector Mi is set equal to the total number of mesh areasand the processes of steps S14 and S15 are completed, the calculationprocess is completed.

In accordance with the simulation program, the value of “j” of themagnetization vector M_(j) included in the magnetic material modeladjacent the magnetization vector M_(i) as a calculation target issuccessively incremented from “0” to the total number of mesh areas-1(the loop between steps S13-1 and S13-2). The timing of incrementing thevalue of “j” corresponds to the timing of S13-2 when the processes ofsteps S14 and S15 are completed, as will be described later.

After step S13-1, the volume dV_(ij) occupied by the two magnetizationvectors M_(i) and M_(j) and the distance L_(ij) between the twomagnetization vectors M_(i) and M_(j) are calculated according toequations (7) and (8). Also, the rotation axis u and the transformmatrix T are calculated according to equations (17) and (18). Further,the rotation angle θ is calculated according to equation (11) (stepS14). Thereafter, the magnetic exchange coupling field H_(ex,ij) iscalculated according to equations (21) and (22) using the calculatedvolume dV_(i,j), distance L_(ij), rotation axis u, transform matrix T,and rotation angle θ, in accordance with the simulation program.

The calculated magnetic exchange coupling field H_(ex,ij) is added tothe preceding total of the magnetic exchange coupling field H_(ex), thusobtaining the total magnetic exchange coupling field H_(ex) (step S15).In step S15, in accordance with the simulation program, the magneticexchange coupling field H_(ex,ij) is calculated each time the values of“i” or “j” of the magnetization vectors M_(i) and M_(j) are incremented.The simulation program may be modified such that in step S15, themagnetic exchange coupling field H_(ex,ij) is calculated according toequation (23).

Thus, in accordance with the simulation program, the rotation anglebetween the two magnetization vectors M_(i) and M_(j) disposed at therespective centers of the adjacent elements used in the finite volumemethod are linearly interpolated with reference to the rotation axis uperpendicular to the two magnetization vectors M_(i) and M_(j), and themagnetic field that acts as a force exerted on the two magnetizationvectors M_(i) and M_(j) is integrated with the linearly interpolatedrotation angle to calculate the magnetic exchange coupling field. Inthis way, the magnetic exchange coupling field can be calculated moreaccurately than is possible with related-art methods.

Preferably, in accordance with the simulation program, the rotationangle between the two magnetization vectors M_(i) and M_(j) areinterpolated, the interval between the two magnetization vectors M_(i)and M_(j) are divided into plural areas, and the magnetic field thatacts as a force exerted on the two magnetization vectors M_(i) and M_(j)are numerically integrated for each divided area in order to calculatethe magnetic exchange coupling field. In this way, the magnetic exchangecoupling field may be calculated more accurately than is possible withrelated-art methods.

Preferably, in accordance with the simulation program, the calculatedmagnetic exchange coupling field may be used as the magnetic exchangecoupling field included in the effective magnetic field of the LLGequation. In this way, highly accurate micro-magnetization analysis maybe realized.

The simulation program used in accordance with an embodiment may includeapplication software for magnetic simulation involvingmicro-magnetization. The micro-magnetization analysis according to anembodiment may be applied to a HDD magnetic head, a MRAM(Magnetoresistive Random Access Memory), or a micro-motor.

EXAMPLE

A micro-magnetization analysis was conducted by using the magneticexchange coupling field calculated by the above method as theexchange-coupling magnetic field of the effective magnetic field of theLLG equation. In this micro-magnetization analysis, under the followingmodel conditions (including magnetic material size and materialcharacteristics), it was determined whether the magnetic domain state issimilar before and after varying the mesh size of the magnetic materialmodel. FIG. 11 illustrates a solution of the magnetic domain state.

(Model Conditions)

Size of magnetic material area (X×Y×Z): 500 nm×500 nm×10 nm

Magnitude of magnetization Ms: 8E+5 (A/m)

Anisotropic magnetic field Ku: 5E+2 (J/m³)

Magnetic exchange coupling coefficient A: 1.3E-11 (J/m)

Attenuation coefficient α: 1

FIGS. 12A through 12C illustrate the results of the micro-magnetizationanalysis in which the magnetic material model was divided by a mesh sizeof 10 nm. FIG. 12A illustrates a magnetization vector state. FIG. 12Billustrates the X-axis direction components of the magnetizationvectors. FIG. 12C illustrates the Y-axis direction components of themagnetization vectors. In FIGS. 12B and 12C, the red portions indicatecomponents of the plus-direction and the blue portions indicatecomponents of the minus-direction. As illustrated, when the magneticmaterial model is divided by the mesh size of 10 nm, a magnetic domainstate similar to the solution illustrated in FIG. 11 is obtained.

FIGS. 13A through 13C illustrate the results of the micro-magnetizationanalysis in which the magnetic material model was divided by the meshsize of 30 nm. FIG. 13A illustrates a magnetization vector state. FIG.13B illustrates the X-axis direction components of the magnetizationvectors. FIG. 13C illustrates the Y-axis direction components of themagnetization vectors. In FIGS. 13B and 13C, the red portions indicatecomponents of the plus-direction, while the blue portions indicatecomponents of the minus-direction. While discontinuities may be seen inthe rotation of the magnetization vectors as illustrated in FIG. 13A, amagnetic domain state similar to the solution of FIG. 11 is obtained asin the case of the result for the mesh size of 10 nm.

Thus, similar magnetic domain states can be obtained even when the meshsize is increased (such as from 10 nm to 30 nm, for example), indicatingthat the above-described method of calculating the magnetic exchangecoupling field is effective. Further, even when the mesh size isincreased (such as from 10 nm to 30 nm, for example), an accuratemicro-magnetization analysis can be performed. When the mesh size istrebled, for example, the degrees of freedom of the LLG equation may bereduced to about 1/27, thus contributing to the decrease (about 1/27) inmemory resource required for calculations. Such an increase in mesh sizealso contributes to the decrease in calculation time (about 1/27).

All examples and conditional language recited herein are intended forpedagogical purposes to aid the reader in understanding the inventionand the concepts contributed by the inventor to furthering the art, andare to be construed as being without limitation to such specificallyrecited examples and conditions, nor does the organization of suchexamples in the specification relate to a showing of the superiority orinferiority of the invention. Although the embodiments of the presentinvention have been described in detail, it should be understood thatthe various changes, substitutions, and alterations could be made heretowithout departing from the spirit and scope of the invention.

What is claimed is:
 1. A non-transitory computer-readable recordingmedium storing a program configured to cause a computer to perform amagnetic exchange coupling energy calculating process, the magneticexchange coupling energy calculating process comprising: linearlyinterpolating a rotation angle between two magnetization vectorsdisposed at the respective centers of two adjacent elements used in afinite volume method, the rotation angle between the two magnetizationvectors being linearly interpolated with reference to a rotation axisthat is shared by the two magnetization vectors and perpendicular to thetwo magnetization vectors that rotate about the rotation axis; andcalculating a magnetic exchange coupling field by integrating a magneticfield acting as a force exerted on the two magnetization vectors withthe linearly interpolated rotation angle.
 2. The non-transitorycomputer-readable medium according to claim 1, wherein the magneticexchange coupling energy calculating process further comprises:calculating the magnetic exchange coupling field in accordance withequations (i) and (ii) obtained by integrating the magnetic field actingas a force exerted on the two magnetization vectors with the linearlyinterpolated rotation angle: $\begin{matrix}{{\overset{->}{H}}_{{ex},{ij}} = {\frac{2A\;{\mathbb{d}S_{ij}}}{M_{S}L\;\Theta_{ij}}{\tau_{ij}\left( \Theta_{ij} \right)}\overset{->}{M_{i}}}} & (i) \\{{\tau_{ij}\left( \Theta_{ij} \right)} = \begin{Bmatrix}{{I_{ij}\left( {{\sin\;\Theta_{ij}} - \Theta_{ij}} \right)} + {\left( {{{- \sin}\;\Theta_{ij}} + \Theta_{ij}} \right)u_{i}u_{j}} +} \\{\left( {1 - {\cos\;\Theta_{ij}}} \right)\begin{pmatrix}0 & {- u_{z}} & u_{y} \\u_{z} & 0 & {- u_{x}} \\{- u_{y}} & u_{x} & 0\end{pmatrix}}\end{Bmatrix}} & ({ii})\end{matrix}$ where H_(ex,ij) is the magnetic exchange coupling field, Ais a magnetic exchange coupling coefficient, dS_(ij) is a sectional areabetween the two elements, M_(s) is a saturation magnetization per unitvolume, L is a distance between the two magnetization vectors, θ_(ij) isthe linearly interpolated rotation angle, I_(ij) is a unit matrix, u_(i)and u_(j) are a rotation axis perpendicular to the two magnetizationvectors, u_(x), u_(y), and u_(z) are vector components of the rotationaxis, and M_(i) is one of the two magnetization vectors.
 3. Thenon-transitory computer-readable medium according to claim 1, whereinthe magnetic exchange coupling energy calculating process furthercomprises: dividing an interval between the two magnetization vectorsinto plural areas; and calculating the magnetic exchange coupling fieldby subjecting the magnetic field acting as a force exerted on the twomagnetization vectors to numerical integration in each of the dividedplural areas.
 4. The non-transitory computer-readable medium accordingto claim 3, wherein the magnetic exchange coupling energy calculatingprocess further comprises: calculating the magnetic exchange couplingfield in accordance with an equation (iii) by which the interval betweenthe two magnetization vectors is divided into plural areas and themagnetic field acting as a force exerted on the two magnetizationvectors is subjected to numerical integration in each of the dividedplural areas: $\begin{matrix}{{\overset{->}{H}}_{{ex},{ij}} = {\frac{2A\;{\mathbb{d}S_{ij}}}{M_{S}L_{ij}N}{\sum\limits_{n = 1}^{N - 1}{w_{i}\left( {{\overset{->}{M}}_{n} - {\overset{->}{M}}_{n + 1}} \right)}}}} & ({iii})\end{matrix}$ where H_(ex,ij) is the magnetic exchange coupling field, Ais a magnetic exchange coupling coefficient, dS_(ij) is a sectional areabetween two elements, M_(s) is a saturation magnetization per unitvolume, L_(ij) is a distance between the two magnetization vectors, N isa number of divisions, w_(i) is a weighting function of a Gaussianintegral, and M_(n) and M_(n+1) are the two adjacent magnetizationvectors.
 5. The non-transitory computer-readable medium according toclaim 1, wherein the program causes the computer to performmicro-magnetization analysis by using the calculated magnetic exchangecoupling field as an exchange-coupling magnetic field included in aneffective magnetic field of a LLG (Landau-Lifshitz-Gilbert) equation. 6.A non-transitory computer-readable recording medium storing a programconfigured to cause a computer to perform a magnetic exchange couplingenergy calculating process, the magnetic exchange coupling energycalculating process comprising: linearly interpolating a rotation anglebetween two magnetization vectors disposed at the respective centers oftwo adjacent elements used in a finite volume method, the rotation anglebetween the two magnetization vectors being linearly interpolated withreference to a rotation axis that is shared by the two magnetizationvectors and perpendicular to the two magnetization vectors that rotateabout the rotation axis; and calculating a magnetic exchange couplingenergy by integrating the energy of a volume occupied by the twomagnetization vectors with the linearly interpolated rotation angle. 7.The non-transitory computer-readable medium according to claim 6,wherein the magnetic exchange coupling energy calculating processfurther comprises: calculating the magnetic exchange coupling energyaccording to an equation (iv) obtained by integrating the energy of thevolume occupied by the two magnetization vectors with the linearlyinterpolated rotation angle: $\begin{matrix}{E_{ij} = {{- \frac{2A\;{\mathbb{d}S_{ij}}}{L_{ij}\Theta}}\left( {\Theta - {\sin\;\Theta}} \right)}} & ({iv})\end{matrix}$ wherein E_(ij) is the magnetic exchange coupling energy, Ais a magnetic exchange coupling coefficient, dS_(ij) is a sectional areabetween the two elements, L_(ij) is a distance between the twomagnetization vectors, and θ is the linearly interpolated rotationangle.
 8. The non-transitory computer-readable medium according to claim6, wherein the magnetic exchange coupling energy calculating processfurther comprises: dividing an interval between the two magnetizationvectors into plural areas; and calculating the magnetic exchangecoupling energy by subjecting the energy of the volume occupied by thetwo magnetization vectors to numerical integration in each of thedivided plural areas.
 9. The non-transitory computer-readable mediumaccording to claim 8, wherein the magnetic exchange coupling energycalculating process further comprises: calculating the magnetic exchangecoupling energy in accordance with an equation (v) by which the intervalbetween the two magnetization vectors is divided into plural areas andthe energy of the volume occupied by the two magnetization vectors issubjected to numerical integration: $\begin{matrix}{E_{ij} = {{- \frac{A\;{\mathbb{d}S_{ij}}}{M_{S}L_{ij}N}}{\sum\limits_{n = 1}^{N}{w_{i}\left( {{\overset{->}{M}}_{n} - {\overset{->}{M}}_{n + 1}} \right)}^{2}}}} & (v)\end{matrix}$ where E_(ij) is the magnetic exchange coupling energy, Ais a magnetic exchange coupling coefficient, dS_(ij) is a sectional areabetween the two elements, M_(s) is a saturation magnetization per unitvolume, L_(ij) is a distance between the two magnetization vectors, N isa number of divisions, w_(i) is a weighting function of a Gaussianintegral, and M_(n) and M_(n+1) are the two adjacent magnetizationvectors.
 10. A magnetic exchange coupling energy calculating methodcomprising: linearly interpolating a rotation angle between twomagnetization vectors disposed at the respective centers of two adjacentelements used in a finite volume method, the rotation angle between thetwo magnetization vectors being linearly interpolated with reference toa rotation axis that is shared by the two magnetization vectors andperpendicular to the two magnetization vectors that rotate about therotation axis; and calculating, using a processor, a magnetic exchangecoupling field by integrating a magnetic field acting as a force exertedon the two magnetization vectors with the linearly interpolated rotationangle.
 11. The magnetic exchange coupling energy calculating methodaccording to claim 10, further comprising: dividing an interval betweenthe two magnetization vectors into plural areas; and calculating themagnetic exchange coupling field by subjecting the magnetic field actingas a force exerted on the two magnetization vectors to numericalintegration in each of the divided plural areas.
 12. The magneticexchange coupling energy calculating method according to claim 10,further comprising: performing micro-magnetization analysis by using thecalculated magnetic exchange coupling field as an exchange-couplingmagnetic field included in an effective magnetic field of a LLG(Landau-Lifshitz-Gilbert) equation.
 13. A magnetic exchange couplingenergy calculating method comprising: linearly interpolating a rotationangle between two magnetization vectors disposed at the respectivecenters of two adjacent elements used in a finite volume method, therotation angle between the two magnetization vectors being linearlyinterpolated with reference to a rotation axis that is shared by the twomagnetization vectors and perpendicular to the two magnetization vectorsthat rotate about the rotation axis; and calculating, using a processor,a magnetic exchange coupling energy by integrating the energy of avolume occupied by the two magnetization vectors with the linearlyinterpolated rotation angle.
 14. The magnetic exchange coupling energycalculating method according to claim 13, further comprising: dividingan interval between the two magnetization vectors into plural areas; andcalculating the magnetic exchange coupling energy by subjecting theenergy of the volume occupied by the two magnetization vectors tonumerical integration in each of the divided plural areas.
 15. Amagnetic exchange coupling energy calculating apparatus comprising: aninterpolating unit configured to linearly interpolate a rotation anglebetween two magnetization vectors disposed at the respective centers oftwo adjacent elements used in a finite volume method, the rotation anglebetween the two magnetization vectors being linearly interpolated withreference to a rotation axis that is shared by the two magnetizationvectors and perpendicular to the two magnetization vectors that rotateabout the rotation axis, and a calculating unit configured to calculatea magnetic exchange coupling field by integrating a magnetic fieldacting as a force exerted on the two magnetization vectors with thelinearly interpolated rotation angle.
 16. The apparatus according toclaim 15, wherein the calculating unit calculates a magnetic exchangecoupling energy by integrating the energy of a volume occupied by thetwo magnetization vectors with the linearly interpolated rotation angle.